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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Hassi, Juuso", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-04-09T05:50:31Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-04-09T05:50:31Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/74990", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on esitt\u00e4\u00e4 vaihtoehtoinen tapa m\u00e4\u00e4ritell\u00e4 jana-aritmetiikka. Yleisesti jana-aritmetiikkaa m\u00e4\u00e4ritelt\u00e4ess\u00e4 on totuttu antamaan janalle pituus, joka usein kiinnitet\u00e4\u00e4n reaalilukuihin. T\u00e4ll\u00e4 tavalla saadaan reaalilukujen algebrallisten ominaisuuksien avulla rakennettua jana-aritmetiikka, mutta t\u00e4ss\u00e4 tutkielmassa l\u00e4hdet\u00e4\u00e4nkin l\u00e4hestym\u00e4\u00e4n janoja t\u00e4ysin geometrian n\u00e4k\u00f6kulmasta. Geometria on yksi matematiikan vanhimmista osa-alueista ja jo noin 300 vuotta ennen ajanlaskun alkua Eukleides Aleksandrialainen julkaisi merkitt\u00e4v\u00e4n teoksen Alkeet. Kirja on yksi kaikkien aikojen menestyneimmist\u00e4 teoksista ja sit\u00e4 k\u00e4ytettiin geometrian oppikirjana yli 2000 vuotta.\n\nGeometrian pohjalle tarvitaan perusoletuksia eli aksioomia, joihin pohjautuen voidaan todistaa tuloksia suoraviivaisesti ja t\u00e4sm\u00e4llisesti. T\u00e4m\u00e4n tutkielman p\u00e4\u00e4l\u00e4hteen\u00e4 on k\u00e4ytetty amerikkalaisen matemaatikon Robin Hartshornen teosta Geometry: Euclid and Beyond. Teoksen aksioomat tulevat saksalaisen matemaatikon David Hilbertin esittelem\u00e4st\u00e4 aksioomaj\u00e4rjestelm\u00e4st\u00e4 tasogeometrialle, jossa h\u00e4n t\u00e4ydensi Eukleideen aiempia aksioomia.\n\nTutkielman alussa m\u00e4\u00e4ritell\u00e4\u00e4n yhteen- ja kertolaskutoimitukset janoille geometrisesti ja n\u00e4iden ominaisuudet todistetaan aksioomien avulla. Kertolaskun ominaisuuksien perusteluissa k\u00e4ytet\u00e4\u00e4n apuna my\u00f6s syklisi\u00e4 nelikulmioita, jotka ovat nelj\u00e4n pisteen joukkoja saman ympyr\u00e4n keh\u00e4lt\u00e4. Nelikulmiot muodostuvat n\u00e4iden pisteiden v\u00e4lisist\u00e4 janoista, jotka eiv\u00e4t leikkaa toisiaan. Laskutoimitusten m\u00e4\u00e4rittelyn j\u00e4lkeen siirryt\u00e4\u00e4n yhdenmuotoisiin kolmioihin, joissa vertaillaan kolmioiden kulmia ja sivujen suhteita. Yhdenmuotoisten kolmioiden avulla voidaan todistaa merkitt\u00e4vi\u00e4 lauseita, kuten Pythagoraan lause.\n \nLopuksi ympyr\u00e4 sulkeutuu osoittamalla, ett\u00e4 geometristen ominaisuuksien pohjalta jana-aritmetiikka toimii samalla tavalla kuin koulumatematiikassa karteesiseen koordinaatistoon ja reaalilukuihin pohjautuva tapa. On siis yht\u00e4pit\u00e4v\u00e4\u00e4 k\u00e4ytt\u00e4\u00e4 molempia geometrian l\u00e4hestymistapoja, mutta geometrisen mallin vahvuus piilee siin\u00e4, ett\u00e4 se ei ole sidottu mihink\u00e4\u00e4n tiettyyn lukuj\u00e4rjestelm\u00e4\u00e4n. N\u00e4iden mallien yht\u00e4l\u00e4isyys todistetaan tutkielmassa tekem\u00e4ll\u00e4 kuvaus geometriselta tasolta karteesiselle koordinaatistolle ja osoittamalla, ett\u00e4 t\u00e4m\u00e4 kuvaus on isomorfinen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-04-09T05:50:31Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-04-09T05:50:31Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "40", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Jana-aritmetiikka geometrisesti", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202104092304", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "aritmetiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "aksioomat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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